My father has been using this phone for almost 6 month snow but recently he acknowledged network problems. his office is in the suburbs where the network is lost...and as soon as he enters the main city the network resumes. What sort of problem this may be?

Network coverage is cell tower and carrier dependent, primarily.
His carrier may have better coverage in the city and sparse outside of the city.
It could also be an issue with the tower near the area he looses coverage, such as hardware or maintenance. It is very expensive to upkeep towers, if the owner is having money issues, they may opt to put off repair or simply let the tower die (like what Sprint did when it took over Nextel)
Other issues can be a "line of sight" thing, such as mountains, valleys, or other very large features that are between the tower and the area he looses coverage.